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The generalized, or conjugate, momentum associated with a coordinate $q_i$ is

In Lagrangian mechanics, the coordinate variable describes the configuration; its conjugate momentum need not equal the Cartesian mechanical momentum.

For electromagnetic scalar and vector potentials $\varphi$ and $A$,Withthe Euler--Lagrange equations give the Lorentz force $m\ddot{r}=q(E+\dot{r}\times B)$, while the canonical momentum is $p=m\dot{r}+qA$.

Replacing a Lagrangian byadds only the endpoint term $F(q(t_2),t_2)-F(q(t_1),t_1)$ to the action. Fixed-endpoint variations therefore give the same Euler--Lagrange equations.

The action of a path is the time integral of its Lagrangian. The principle of stationary action requires its first variation to vanish for every admissible fixed-endpoint variation.

For a topless rectangular box with dimensions $x,y,z$, volume $V$, and weighted face costan interior optimum equalizes the three terms:The same result and global minimality follow from the arithmetic-geometric mean inequality.

Forsuppose the optimizer and multiplier vary smoothly and use $L=f-\lambda (g-b)$. Differentiating the optimum value and using stationarity gives

For a jump process with rate $W(n,r)$ from state $n$ to $n+r$, probability balance givesThe first term is inflow from $n-r$ and the second is outflow from $n$.

For jumps $n\to n+1$ at rate $\lambda$ and $n\to n-2$ at rate $\beta n^2$, the diffusion approximation hasThe stable deterministic population is $n_{\ast }=\sqrt{\lambda /(2\beta )}$. Its linear-noise stationary approximation is normal with mean $n_{\ast }$ and variance $3n_{\ast }/4$.

A soliton is a localized travelling wave whose shape is preserved by the evolution and whose interactions with other solitons are elastic up to phase shifts.

The inverse-scattering spectral problem associated with this KdV convention isFordifferentiation gives the Wronskian identity

A colored link assigns each component a color. In the trivial coloring all components have the same color, so a doubly slice realization uses one unknotted surface component.

A link is a smooth embedding of a finite disjoint union of circles in $S^3$, considered up to ambient isotopy. A one-component link is a knot.

The Seifert longitude is the zero-linking parallel of a knot on the boundary of its tubular neighborhood. It is the framing induced by a Seifert surface.

The Stevedore knot $6_1$ is a ribbon knot with Alexander polynomial $2t^2-5t+2$ up to a unit.

A surface knot is a smooth embedding of a closed connected surface in a four-manifold. In the narrow standard usage, a 2-knot is an embedded two-sphere in $S^4$.

It extends the embedding viewpoint of knot theory to two-dimensional domains.

An $m$-twist spin removes a trivial arc from a knot and spins the resulting knotted arc around the boundary of a three-ball in $S^4$, inserting $m$ full twists during one revolution.

The result is a surface knot with domain a two-sphere.

The Arf invariant is a $\mathbb{Z}/2$-valued concordance invariant obtained from the quadratic refinement of the mod-two intersection form on a Seifert surface.

An amphichiral knot is ambient-isotopic to its mirror. Positive and negative amphichirality distinguish whether the symmetry preserves or reverses the knot orientation.

The two-fold branched cover of a knot is the double cover of $S^3$ branched along the knot. Its first homology has a nonsingular linking form and, when finite, has order $\det K$.

Away from the knot this is a two-sheeted covering space; at the knot it has the standard local model of a branched covering.

Two knots are concordant when they cobound a smoothly embedded annulus in $S^3\times [0,1]$. Connected sum makes concordance classes into an abelian group.